$$ \int_{-\infty}^{+\infty}{f(t)\psi_{j,k}^\ast(t)dt}\ \textrm{with}\ \psi_{j,k}(t)\ =\ a_0^{-j/2}\psi(a_0^{-j}t\ -\ b_0k) $$ If this is the expression for the wavelet transform, how does this lead to wavelet decomposition and at what stage do we compute approximations and details?

  • $\begingroup$ i might guess that you actually mean: $$ \psi_{j,k}(t)\ =\ a_0^{-j/2}(a_0^{-j(t-b_0k)}u(t-b_0k)) $$ (where $u(t)$ is the Heaviside unit step function) rather than the expression you have. $\endgroup$ – robert bristow-johnson Nov 23 '19 at 22:09

This expression is more a discretization of a continuous wavelet transform than an actual DWT (discrete wavelet transform), provided $\psi$ is a genuine wavelet. It only computes the wavelet coefficient $c_{j,k}$ associated to a specific shifted and dilated continuous wavelet $\psi_{j,k}(t)$. This yields a frame-like wavelet decomposition, if you picture all possible $c_{j,k}$. A frame is a kind of overcomplete basis satisfying some bounds; in other words, more projection vectors than the dimension of the space, still ensuring a certain stability.

It is not a discrete wavelet transform per se, since time is still continuous. This is a bit like Fourier series. Approximations and details for DWT are more classically obtained from a direct discrete scheme. From a discrete signal $x[n]$, at one single level, it is possible to build two downsampled sequences:

  • $a(n) = (h_0\ast x)\downarrow(2)[n]$ (approximation)
  • $d(n) = (h_1\ast x)\downarrow(2)[n]$ (details)

with $h_0$ and $h_1$ complementary (low-pass and high-pass) filters, such that one can recover $x$ from half-length sequences $a(n)$ and $b(n)$ only. Coefficients for $h_0$ and $h_1$ can be obtained from certain wavelet shapes only. Passing the signal in a filter before downsampling reduces aliasing artifacts, while preserving invertibility.

Approximations and details (discrete wavelets and wavelet packets) can be obtained by iterating the above at different levels. Stationary wavelets are a version of DWT without subsampling (but generally the same wavelets). They often behave better in case of higher noise,for denoising, or model mismatch in adaptive filtering (some details in A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal).

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    $\begingroup$ What exactly do you mean by a frame-like wavelet decomposition? One more thing... Is there any specific reason as to why we down sample after passing the signal through each filter? And if there is, then why does the stationary wavelet transform exist and where do we use it? $\endgroup$ – Dhanush Giriyan Oct 26 '19 at 6:38
  • $\begingroup$ A frame is a kind of overcomplete basis satisfying some bounds. Passing the signal in a filter before downsampling reduces aliasing artifacts. Stationary wavelets behave better in case of noise or model mismatch $\endgroup$ – Laurent Duval Nov 24 '19 at 12:11

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